If `f(x) = 3x-5`, then `f^(-1) (x)`

Options

A. is given by `1/(3x-5)`
B. is given by `(x+5)/3`
C. does not exist because f is not one-one
D. does not exist because f is not onto.

Answer

To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = 3x - 5 \), we will follow these steps:

### Step 1: Set \( f(x) \) equal to \( y \)
Let \( y = f(x) \).
\[ y = 3x - 5 \]

### Step 2: Solve for \( x \) in terms of \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \). Start by isolating \( x \):
\[ y + 5 = 3x \]
Now, divide both sides by 3:
\[ x = \frac{y + 5}{3} \]

### Step 3: Write the inverse function
Now that we have \( x \) in terms of \( y \), we can express the inverse function:
\[ f^{-1}(y) = \frac{y + 5}{3} \]

### Step 4: Replace \( y \) with \( x \) to express \( f^{-1}(x) \)
To write the inverse function in terms of \( x \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{x + 5}{3} \]

### Final Answer
Thus, the inverse function is:
\[ f^{-1}(x) = \frac{x + 5}{3} \]

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